Macroscopic fluids with microscopic configuration/structure Competition between the kinetic energy and the internal “elastic” energy Intrinsically multiscale-multiphysics model, or multicomponent

Maximum dissipation principle in numerical simulation of

complex fluids

Yunkyong Hyon

Institute for Mathematics and Its Applications

IMA PI Workshop on Numerical Simulation of Complex Fluids and MHD, Penn. State University

Outline

Complex Fluids

Complex Fluids

Macroscopic fluids with microscopic configuration/structure

Competition between the kinetic energy and the internal

“elastic” energy

Complex Fluids

Macroscopic fluids with microscopic configuration/structure

Competition between the kinetic energy and the internal

“elastic” energy

Complex Fluids

Macroscopic fluids with microscopic configuration/structure

Competition between the kinetic energy and the internal

“elastic” energy

Complex Fluids : Examples

Viscoelastic Fluids

(

)

Electrorheological (ER) Fluids

(

)

Complex Fluids

The analysis of complex fluids is intrinsically difficult.

Numerical computations are highly demanded to understand the

complex fluids.

The starting point of energetic variational approaches: Energy

dissipation law for the whole coupled system,

d

dt

E

total

₌

−△

where

E

_{is the total energy,}

_△

_{is the dissipation of the}

Complex Fluids

The analysis of complex fluids is intrinsically difficult.

Numerical computations are highly demanded to understand the

complex fluids.

The starting point of energetic variational approaches: Energy

dissipation law for the whole coupled system,

d

dt

E

total

₌

−△

where

E

_{is the total energy,}

_△

_{is the dissipation of the}

Energetic Variational Approaches: Flow Map

b b

~x(X~,t) Ω0

~ X

Ωt

~x

Figure: The flow map from the reference domain, Ω0to the current domain, Ωt.

Basic Mechanics

Flow Map (Trajectory) : ~xt(X~,t) =~u(~x(X~,t),t), ~x(X~,0) =X~.

The deformation tensor (strain) of the flow map :

F(~x(X~,t),t) = ∂~x(X~,t)

∂ ~X satisfying

Energetic Variational Approaches

Least Action Principle (Hamiltons Principle; Principle of Virtual

Work) :

the reversible part of the system, conservative force (

f

)

δ

_A

=

f

·

δ~

x

where

_A

is the action functional defined from the energy.

Maximum Dissipation Principle

(Onsager’s Principle) :

the

irreversible part of the system, the dissipative force (

f

)

1

2

δ

△

=

f

·

δ~

u

.

Energetic Variational Approaches : Simple Fluid

Simple Fluid :

the fluid described by the incompressible Navier-Stokes

equations

ρ

(

~

u

+

~

u

· ∇

~

u) +

∇

p

=

µ

∆

~

u

,

(momentum conservation)

∇ ·

~

u

= 0

,

(incompressibility)

(2)

ρ

+

∇ ·

(

ρ~

u

) = 0

.

(mass conservation)

where

ρ

is mass,

~

u

is velocity field,

p

is pressure, and

µ

is the viscosity.

The energy equation for incompressible Navier-Stokes equation:

d

dt

Z

1

2

ρ

|

~

u

|

2

_d

_~

_x

₌

₋

Z

Energetic Variational Approaches : Simple Fluid

The total energy, the dissipation :

E

=

Z

1

2

ρ

|

~

u

|

2

_d

_~

_x

_,

△

=

Z

Energetic Variational Approaches : Simple Fluid

Define Action Functional,

_A

, for

Least Action Principle (LAP)

A

=

Z

0

Z

Ωt

1

2

ρ

|

~

u

|

2

_d

_~

_xdt

_.

₍₅₎

Pull back the current domain, Ω

, to the reference one, Ω

, through

the flow map,

~

x

(

X

~

,

t

)

A

(

~

x) =

Z

0

Z

Ω0

1

2

ρ

(

X

~

)

det

F

|

~

x

|

2

_det

_{F d}

_{X dt}

_~

₍₆₎

where

ρ

(

X

~

) =

ρ

(

X

~

,

t

)

|

is the initial mass. Then

δ~x

= 0 gives

the Hamiltonian (energy conserved) part under the incompressibility

condition,

_{∇ ·}

~

u

= 0, i.e., det(

F

) = 1.

The resulting equation is the Euler equation : the total energy

conservation

ρ

(

~

u

+

~

u

· ∇

~

u) =

−∇

p

Energetic Variational Approaches : Simple Fluid

Applying

Maximum Dissipation Principle (MDP)

for the dissipation

in the form of the quadratic of the “rate” functions,

1

2

δ

_△

δ~

u

=0

=

Z

(

_∇

~

u

+

ε

_∇

~

v

)

_{· ∇}

~

v d

~

x

=0

= 0

,

then we obtain the Stokes equation,

µ

∆

~

u

=

_∇

˜

p

,

(8)

∇ ·

~

u

= 0

.

Energetic Variational Approaches : Framework

1

Define an appropriate energy equation for describing the physical

phenomenon.

2

Applying energetic variational approaches, LAP, MDP.

3

Combine the conservative part from LAP and the dissipative part

from MDP through the force balance to satisfy the energy law.(The

system of PDE is uniquely determined.)

4

Numerical Computations, developing numerical methods, existence,

Energetic Variational Approaches : Framework

1

Define an appropriate energy equation for describing the physical

phenomenon.

2

Applying energetic variational approaches, LAP, MDP.

3

Combine the conservative part from LAP and the dissipative part

from MDP through the force balance to satisfy the energy law.(The

system of PDE is uniquely determined.)

4

Numerical Computations, developing numerical methods, existence,

Energetic Variational Approaches : Framework

1

Define an appropriate energy equation for describing the physical

phenomenon.

2

Applying energetic variational approaches, LAP, MDP.

3

Combine the conservative part from LAP and the dissipative part

from MDP through the force balance to satisfy the energy law.(The

system of PDE is uniquely determined.)

4

Numerical Computations, developing numerical methods, existence,

Energetic Variational Approaches : Framework

1

Define an appropriate energy equation for describing the physical

phenomenon.

2

Applying energetic variational approaches, LAP, MDP.

3

Combine the conservative part from LAP and the dissipative part

from MDP through the force balance to satisfy the energy law.(The

system of PDE is uniquely determined.)

4

Numerical Computations, developing numerical methods, existence,

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow

:

Letφbe a phase functionφ(~x,t) =±1 in the incompressible fluids

Let Γt ={x:φ(~x,t) = 0}be the interface of mixture

In the Eulerian description, the immiscibility of fluids implies the pure transport ofφ.

φt+~u· ∇φ= 0. (9)

2

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow:

Ginzburg-Landau mixing energy, represents a competition between two fluids with their (hydro-) philic and (hydro-) phobic properties:

W(φ,∇φ) = 1 2|∇φ|

2₊ 1

4η2(φ 2₋₁₎2_.

The equilibrium profile of interface is tanh-like function asη→0.

The total energy is defined by the combination of the kinetic energy and the internal energy as follows:

Etotal =

Z

Ωt

₁

2|~u|

2₊

λW(φ,∇φ)

d~x. (10)

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow:

The action functionalAfrom the total energy (10):

A= Z T 0 Z Ωt 1 2|~xt|

2

−λW(φ,_∇φ)

d~xdt. (11)

The action functionalAin terms of the flow map:

A(~x) =

Z T 0 Z Ω0 1 2|~xt|

2 −λ

1 2|F

−1_∇ ~ Xφ|

2_+f(φ0)

detF

dX dt.~

(12) wheref(φ0) = 1

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow without Dissipation:

the LAP leads us to the following Hamiltonian system:

~ut+~u· ∇~u+∇p˜ = −λ∇ ·(∇φ⊗ ∇φ) (13)

∇ ·~u = 0 (14)

φt+~u· ∇φ = 0. (15)

The above system (18) – (20) converges to the “sharp interface model” and its energy equation is

d dt

Z ₁

2|~u|

2_{+λ 1}

2|∇φ|

2₊ 1

4η2(φ 2

−1)2

d~x = 0. (16)

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow:

Consider the following dissipation with the viscosity of fluids and the dissipation within the diffusive interface:

△=

Z

µ|∇~u|2+λγ

∆φ− 1

η2(φ 2

−1)φ

2!

d~x. (17)

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow:

The system with dissipation:

~ut+~u· ∇~u+∇p˜ = µ∆~u−λ∇ ·(∇φ⊗ ∇φ) (18)

∇ ·~u = 0 (19)

φt+~u· ∇φ = γ

∆φ₋ 1 η2 φ

2 −1

φ

. (20)

The dissipative energy law is

d dt

Z

1 2|~u|

2_+λ

1 2|∇φ|

2₊ 1

4η2(φ 2

−1)2

d~x

(21) =−

Z

µ|∇~u|2+λγ

∆φ− 1 η2(φ

2 −1)φ

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow with Dissipation:

Reducing the computational cost in numerical experiment through the

Maximum dissipation principle.

Manipulate the dissipation (17) in terms of a rate

△=

Z

µ|∇~u|2+λ

γ|φt+~u· ∇φ| 2_d~

x. (22)

Onsarger’s principle (MDP) with incompressibility of flow,∇ ·~u= 0 implies

1 2

δ_△ δ~u

ε=0

=− Z

µ∆~u₋λ

γ(φt+~u· ∇φ)∇φ

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow:

The resulting system obtained by LAP and MDP with the manipulation on the dissipation is

~ut+~u· ∇~u+∇p = µ∆~u−

λ

γ(φt+~u· ∇φ)∇φ (24)

∇ ·~u = 0 (25)

φt+~u· ∇φ = γ

∆φ₋ 1 η2(φ

2 −1)φ

. (26)

The dissipative energy law:

d dt

Z

1 2|~u|

2_+λ

1 2|∇φ|

2₊ 1

4η2(φ 2

−1)2

d~x

(27)

Z

Energetic Variational Approaches : Complex Fluid

Remark:

The dissipative force term in (24),

−λ

γ(φt+~u· ∇φ)∇φ

is exactly same as the conservative force in (18),

∇ ·(∇φ⊗ ∇φ−W(φ,∇φ)I)

Asη _→0, this is exactly the surface tension force on the interface3

3

Energetic Variational Approaches : Complex Fluid

Immiscible Two-phase Flow (continue):

Computational Viewpoints:

d dt

Z ₁

2|~u|

2_{+λ 1}

2|∇φ|

2₊ 1

4η2(φ 2₋₁₎2

d~x

(28) =−

Z

µ|∇~u|2+λ

γ|φt+~u· ∇φ| 2

d~x.

: Lower Order Approximation in Numerical Simulations

d dt

Z ₁

2|~u|

2₊ λ

₁

2|∇φ|

2₊ 1

4η2(φ 2

−1)2

d~x

(29) =−

Z

µ|∇~u|2+λγ|∆φ− 1

η2(φ 2

−1)φ|2

Energetic Variational Approaches : Complex Fluid

Another Example:

an incompressible viscoelastic complex fluid

model

with the elastic energy,

W

(

F

) =

_|

F

_|

The energy law:

d dt

Z

1 2|~u|

2₊1

2|F|

2

d~x=− Z

µ_|∇~u_|2d~x. (30)

The system of equations satisfies the energy law (30):

~ut+~u· ∇~u+∇p = µ∆~u+∇ · WFF−T (31)

∇ ·~u = 0 (32)

Ft+~u· ∇F = ∇~uF (33) : WFF−T is conservative force.

4

Energetic Variational Approaches : Complex Fluid

Another Example (continue):

Add an artificial term ∆

F

into (33)

F

+

~

u

· ∇

F

=

∇

~

uF

+

ε

∆

F

.

(34)

The energy law of the system (31), (32), (34):

d dt

Z ₁

2|~u|

2₊1

2|F|

2

d~x=− Z

µ|∇~u|2_+ε2_|∇F|2

d~x. (35)

Apply MDP: the resulting energy equation is

d dt

Z ₁

2|~u|

2₊1

2|F|

2 d~x

(36) =−

Z

Numerical Simulations

Finite Element Space

:

~

u

∈

V

= (P

⊕

bubble)

(37)

p

∈

W

=

P

(38)

φ

∈

Q

=

P

(39)

Numerical Simulations

Numerical Method : The explicit-implicit scheme

(˜~u_h,tn+1, ~vh) +

_3~

u_hn−~un−1 h

2 · ∇

~u

n+1 2

h , ~vh

+

₁

2

∇ ·3~u

n h−~u

n−1 h 2 ~u n+1 2

h , ~vh

−(p

n+1 2

h ,∇ ·~vh) =−(µ∇~u

n+1 2

h ,∇~vh)− λ γ

˜

φn_h,t+1∇ _3φn

h−φ n−1 h

2

, ~vh

(40) −λ γ ~ un+12

h ,∇

3φn h−φn−

1 h

2 ∇

3φn h−φn−

1 h

2

, ~vh

for all~vh∈Vh,

∇ ·~un+12

h ,wh

+εpn+12

h ,wh

= 0 for allwh∈Wh, (41)

( ˜φn_h,t+1,qh) +

~ u n+1 2 h ,

_3φn h−φ

n−1 h 2 qh (42) =−γ∇φ

n+1 2

h ,∇qh

− γ

η2(fh(φ n

Numerical Simulations

where

fh(φnh, φn +1 h ) =

₍

|φn_h+1_|2_{−1) + (|φ}n h|2−1)

2 φ n+1 2 h , ˜ ~

u_h,tn+1=~u

n+1 h −~unh

∆t , ~u

n+1 2

h =

~u_hn+1+~un h

2 ,φ˜

n+1 h,t =

φn_h+1−φn_h

∆t , φ

n+1 2

h =

φn_h+1+φn_h

2 , (·,·) is the inner product operator, and ∆t is the time step for simulations andε

is a small positive constant, 0< ε <<1. The scheme satisfies the energy dissipation law:

Z ₁

2|~u

n+1 h |

2₊ λ

₁

2|∇φ

n+1 h |

2₊ 1

4η2|φ n+1 h

2 −1|2

d~x h,t = (43) − Z (

µ_|∇~un_h+1_|2+ε_|φn +1 2

h | 2₊λ

γ ˜

φn_h,t+1+ (~un +1 2

h · ∇)

3φn h−φn−

Numerical Simulations

Simulation 1 - Initial Data

:

~

u

=

~

0

,

φ

= tanh

_d

1

(x

,

y

)

η

√

2

+ tanh

_d

2

(x

,

y

)

η

√

2

−

1

.

0

(44)

with

d

(x

,

y

) =

q

(x

₋

0

.

38)

_{+ (y}

₋

₀

_.

₅₎

₋

₀

_.

₁₁

_,

d

(x

,

y

) =

q

Numerical Simulations

Simulation 1.- Initial Data,

φ

:

Numerical Simulations

Plot of Interfaces and Flow Fields at Time =0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7

0.8 Plot of Interfaces and Flow Fields at Time =0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7

0.8 Plot of Interfaces and Flow Fields at Time =0.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Plot of Interfaces and Flow Fields at Time =0.4

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Plot of Interfaces and Flow Fields at Time =0.7

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7

0.8 Plot of Interfaces and Flow Fields at Time =1

Numerical Simulations

0 0.2 0.4 0.6 0.8 1 1.2 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014

Time : t

Energy

Energy Dissipation : Merging Effect

Total Energy Mixing Energy Kinetic Energy

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10

−6

Time : t

Energy

Kinetic Energy Evolution : Merging Effect

Numerical Simulations

Simulation 2 - Initial Data

:

~

u

=

~

0

,

φ

= tanh

_d

1

(x

,

y

)

η

√

2

+ tanh

_d

2

(x

,

y

)

η

√

2

−

1

.

0

(45)

with

d

(x

,

y) =

q

(x

₋

0

.

38)

_{+ (y}

₋

₀

_.

₃₈₎

₋

₀

_.

₂₂

_,

d

(x

,

y) =

q

Numerical Simulations

Simulation 2.- Initial Data,

φ

:

Numerical Simulations

Plot of Interfaces and Flow Fields at Time =0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.9 Plot of Interfaces and Flow Fields at Time =0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Plot of Interfaces and Flow Fields at Time =0.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.9 Plot of Interfaces and Flow Fields at Time =2

Numerical Simulations

0 0.5 1 1.5 2 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Time : t

Energy

Energy Dissipation : Vanishing Effect

Total Energy Mixing Energy Kinetic Energy

0 0.5 1 1.5 2 0

0.5 1 1.5 2 2.5 3x 10

−6

Time : t

Energy

Kinetic Energy Evolution : Vanishing Effect

Remarks

LAP : Conservative Force

MDP : Dissipative Force

MDP : Whether the dissipation functional includes the “rate”

functions in time

t

of all “variables”

The conservative force in hydrodynamic system is consistent with the

dissipative force

MDP plays an important role in designing numerical algorithms to

solve the hydrodynamic complex fluid problem

Collaborators

Chun Lin

: IMA Associate Director/ Dept. Mathematics at

Pennsylvania State University

Qiang Du

: Dept. Mathematics/Material science at Pennsylvania

State University

Bob Eisenberg

: Division of Molecular Biophysics and Physiology at

Rush University

Tai-chia Lin

: Dept. Mathematics at National Taiwan University

Jiangfang Huang

: Dept. Mathematics at University of North

Carolina, Chapel Hill

Chiun-Chang Lee

: Dept. Mathematics at National Taiwan University

Huan Sun

: Dept. Mathematics at Pennsylvania State University